# Relating the holomorphic Euler characteristic of a family of algebraic varieties to properties of the base and fibers

Let $$f : X\rightarrow Y$$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $$Y$$ over $$\mathbb{C}$$. Let $$X_{y_0}$$ denote a smooth fiber over $$y_0\in Y$$.

If $$f$$ is smooth, then the topological Euler characteristic is multiplicative: $$\chi(X) = \chi(Y)\cdot\chi(X_{y_0})$$. In general, there is a formula for the topological Euler characteristic which includes some correction terms for the singular fibers: $$\chi(X) = \chi(Y)\chi(X_{y_0}) + \sum_{y\in Y}(\chi(X_y) - \chi(X_{y_0})$$

Does there exist an analogous formula for the holomorphic/algebraic Euler characteristic? I'm in particular interested in the case where $$X\rightarrow Y$$ is a family of curves over $$Y$$ (so $$\dim X = 2$$). To be precise, I'm looking for a formula for $$\chi(X,\mathcal{O}_X)$$ in terms of geometric invariants of $$Y$$ and some fibral data. By fibral data I mean geometric properties which are local on the base $$Y$$ (e.g. geometric invariants of fibers, monodromy around singular fibers acting on the cohomology of a smooth fiber...etc).

As a nonexample, Riemann-Roch relates this Euler characteristic to the intersection number of $$X_{y_0}$$ with $$K - X_{y_0}$$ (which can be understood locally on $$Y$$) plus $$\chi(X,\mathcal{O}_X(X_{y_0}))$$ (which does not appear to be local on $$Y$$).

Added in edit: To state a more precise question -- Is the holomorphic Euler characteristic $$\chi(X,\mathcal{O}_X)$$ determined by $$Y$$ and data which is local on $$Y$$? Stated another way, do there exist maps $$f : X\rightarrow Y$$ and $$f' : X'\rightarrow Y$$ (satisfying the above conditions) such that

1. there is a covering in the etale topology $$\{U_i\rightarrow Y\}_{i\in I}$$ and isomorphisms $$\phi_i : X_{U_i}\rightarrow X'_{U_i}$$, and
2. $$\chi(X,\mathcal{O}_X)\ne \chi(X',\mathcal{O}_{X'})$$.

Here I'm happy to assume that $$X,X'$$ are smooth over $$\mathbb{C}$$ (but I still want to allow singular fibers of $$f,f'$$). If it changes the answer, I'm also interested in the variation where $$\{U_i\}$$ is instead an open covering in the analytic topology. I am also interested in results where we assume that $$X,X'$$ are moreover general type. If the euler characteristic is not determined by $$Y$$ and data local on $$Y$$, then I'm also interested in any contraints such data might impose on $$\chi(X,\mathcal{O}_X)$$.

• I am trying to understand how your "nonexample" is not an example. What kind of formula do you envision when $X$ equals $Y\times Z$ if you are not allowing $\chi(Z,\mathcal{O}_Z)$ as part of your formula? Apr 9, 2022 at 0:31
• Are you willing to assume that $X$ (not $f$) is smooth? Apr 9, 2022 at 7:42
• @JasonStarr I imagine $\chi(X_{y_0},\mathcal O_{X_{y_0}})$ would be allowed as it is 'fibral', but not $\chi(X,\mathcal O_X(X_{y_0}))$ as it is 'global' on $X$. Apr 9, 2022 at 14:12
• @JasonStarr Remy is right - $\chi(X_{y_0},\mathcal{O}_{X_{y_0}})$ would be fine, but $\chi(X, \mathcal{O}_X(X_{y_0}))$ wouldn't be, unless it can somehow be expressed in terms of data which is local on $Y$. Apr 9, 2022 at 15:09
• @PiotrAchinger Sure, I'd be happy for an answer in the case of smooth $X$. Apr 10, 2022 at 20:07

I'm not entirely sure what would constitute an answer. But here a few simple observations. Let me focus on what you seem be interested in, namely a projective family of connected curves over smooth projective curve $$f:X\to Y$$. Let me also assume $$X$$ smooth. Let $$h$$ be the genus of $$Y$$, and $$g$$ the genus of the general fibre. Then from the Leray spectral sequence, Riemann-Roch and Grothendieck duality $$\chi(\mathcal{O}_X)= (1-g)(1-h) - \deg R^1f_*\mathcal{O}_X=\chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})+\deg f_*\omega_{X/Y}$$ A theorem of Fujita then gives an inequality $$\chi(\mathcal{O}_X)\ge \chi(\mathcal{O}_Y)\chi(\mathcal{O}_{X_{y_0}})$$ which is something. To say more, one would need to compute $$\deg f_*\omega_{X/Y}$$. But I think this depends on more than local information at the bad fibres, which is what I think you are asking. The reason I say this, is that this degree can be positive and variable for Kodaira surfaces, which have no bad fibres at all. [Added Comment I guess there are various things called Kodaira surfaces. The ones I have in mind are of general type, and hence algebraic. See page 220 of Compact Complex Surfaces by Barth, Hulek, Peters and Van de Ven for further details.]
• Somehow the point is that the Euler characteristic of a rank $n$ vector bundle is not just $n\chi(X,\mathcal O_X)$, but also depends on global data like the degree. By contrast (and somewhat to my surprise), for local systems the situation is much easier: $\chi(X,\mathcal L) = \operatorname{rk}(\mathscr L)\chi(X,\underline{\mathbf Q})$. Apr 10, 2022 at 23:29